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Introductory Physics Homework Help
Normalising superposition of momentum eigenfunctions
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[QUOTE="alec_grunn, post: 5506918, member: 568380"] Hi Simon, thanks for the response. I've just realized the second option I gave "The allowed momentum values are not p=±ℏk, but ## p = ± \frac{\hbar k}{5A2}##" is not implied by anything above - please forget about it. First off, I forgot to mention it's not in a potential well. So this is for all real values of x. So my question is, doesn't this equation ⟨p⟩=∑ℏkC[SUB]n[/SUB]^2 imply ∑C[SUB]n[/SUB]^2 = 1, since the textbook also says that each C[SUB]n[/SUB][SUP]2[/SUP] is the probability of observing the corresponding momentum value? And therefore we can normalise any linear combination of momentum eigenfunctions, even Ψ=Aexp(-ikx). But that makes me uneasy because (a) it clearly doesn't meet the normalisation condition, and (b) I've read elsewhere that plane waves can't be normalised, for instance [URL='http://physics.stackexchange.com/questions/165373/normalizing-the-solution-to-free-particle-schr%C3%B6dinger-equation']here.[/URL] Maybe it's got something to do with my definition of 'normalisation'. I just read [URL='https://www.physicsforums.com/threads/normalizing-a-wave-function-what-does-it-mean.475368/']this post [/URL]which basically says normalisation is strictly defined as making your wavefunction satisfy ∫ψ⋆ψ=1 over all x values. If so, then the wavefunction I was working with can't be normalised. If it just means something like 'fixing the constant outside the brackets', then I can normalise this wavefunction. Also, apologies for the poor formatting - I'm still coming to terms with using Latex. [/QUOTE]
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Normalising superposition of momentum eigenfunctions
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